Buckling Of Long Compression-Loaded Anisotropic Plates Restrained Against Inplane Lateral and Shear Deformations

37 Pages·2003·0.99 MB·English

by  Nemeth, Michael P

AIAA-2003-1789
BUCKLING OF LONG COMPRESSION-LOADED ANISOTROPIC PLATES
RESTRAINED AGAINST INPLANE LATERAL AND SHEAR DEFORMATIONS
Michael P. Nemeth*
Mechanics and Durability Branch, NASA Langley Research Center
Hampton, Virginia 23681-2199
Abstract sential to the advancement of structural-tailoring
technology for aerospace vehicles.
An approach for synthesizing buckling results and
behavior for thin balanced and unbalanced symmetric An important type of long plate that appears as an
laminates that are subjected to uniform axial compression element of advanced composite structures is the symmet-
loads and elastically restrained against inplane expansion, rically laminated plate. In the present study, the term,
contraction, and shear deformation is presented. This ap- "symmetrically laminated," refers to plates in which ev-
proach uses a nondimensional analysis for infinitely long, ery lamina above the plate midplane has a corresponding
flexurally anisotropic plates (coupling between bending lamina located at the same distance below the plate mid-
and twisting) that are subjected to combined mechanical plane, with the same thickness, material properties, and
loads and is based on nondimensional parameters. In ad- fiber orientation. Symmetrically laminated plates are,
dition, nondimensional loading parameters are derived for the most part, flat after the manufacturing process and
that account for the effects of the elastic inplane deforma- exhibit flat prebuckling deformation states, which is de-
tion restraints, membrane orthotropy, and membrane sirable for many applications. More importantly, the
anisotropy on the induced prebuckling stress state. The amenability of these plates to structural tailoring pro-
loading parameters are used to determine buckling coeffi- vides symmetrically laminated plates with a significant
cients that include the effects of flexural orthotropy and potential for reducing the weight of aerospace vehicles or
flexural anisotropy. Many results are presented, for some for meeting special performance requirements. Thus,
selected laminates, that are intended to facilitate a struc- understanding the buckling behavior of symmetrically
tural designer’s transition to the use of the generic buck- laminated plates is an important part of the search for
ling design curves that are presented and discussed in the ways to exploit plate orthotropy and anisotropy to reduce
paper. Several buckling response curves are presented structural weight or to fulfill a special design require-
that provide physical insight into the behavior for com- ment.
bined loads, in addition to providing useful design data. In many practical cases, symmetrically laminated
An example is presented that demonstrates the use of the plates exhibit specially orthotropic behavior. However,
generic design curves, which are applicable to a wide in many cases, these plates exhibit anisotropy in the form
range of laminate constructions. The analysis approach of material-induced coupling between pure bending and
and generic results indicate the effects and characteristics twisting deformations. This coupling is referred to here-
of laminate orthotropy and anisotropy in a very general in as flexural anisotropy and it generally yields buckling
and unifying manner. modes that are skewed in appearance (see Fig. 1). Sym-
metrically laminated plates that are unbalanced are also
being investigated for special-purpose uses in aerospace
Introduction
structures. In the present study, the term, "unbalanced
Structural tailoring of laminated-composite plates to laminate," refers to symmetric laminates in which each
enhance their buckling resistance is an important element ply with a positive-valued fiber orientation is not “bal-
in the development of new, advanced aerospace vehicles. anced” by a corresponding ply with a negative-valued fi-
One structural component that is often examined in design ber orientation. Unbalanced laminated plates exhibit
of aircraft and spacecraft is the long rectangular plate. anisotropy in the form of material-induced coupling be-
Plates of this type commonly appear as elements of stiff- tween pure inplane dilatation and inplane shear deforma-
ened panels that are used for wing structures, and as semi- tions, in addition to flexural anisotropy. This coupling is
monocoque shell segments that are used for fuselage and referred to herein as membrane anisotropy and it gener-
launch vehicle structures. Thus, establishing a broad un- ally yields combined inplane stress states for simple
derstanding of the buckling behavior of long plates is es- loadings like uniform edge compression when inplane
displacement constraints are imposed on one or more
* Assistant Head. Associate Fellow, AIAA.
edges of a plate. For example, when the two unloaded,
Copyright  2003 by the American Institute of Aeronautics and Astronautics, Inc. No copyright opposite edges of an unbalanced, symmetrically laminat-
is asserted in the United States under Title 17, U. S. Code. The U. S. Government has a royalty-
free license to exercise all rights under the copyright claimed herein for Governmental purposes. ed plate that is compression loaded, such as a [+45/0/
All other rights are reserved by the copyright owner. 2
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90] laminate, are totally restrained against expansion a typical graphite-epoxy material, for several plate aspect
s
and contraction and inplane shearing deformations, in- ratios and five different boundary conditions. Bedair13, 14
plane shear stresses are developed in addition to the bi- and Walker, et. al.15, have also presented studies that in-
axial compression stresses that are typically present in clude the effects of restrained inplane movement, along
balanced laminates (see Fig. 2). These kinematically in- with the presence of nonuniform applied compression
duced shear stresses can be relatively large in magnitude, loads. Specifically, the results in Refs. 13 and 14 focus
compared to the direct compressive stresses, and as a re- on the behavior of finite-length isotropic plates with elas-
sult can affect greatly the buckling behavior of the plate tic inplane restraints. The results presented in Ref. 15 fo-
and yield buckling modes that are skewed in appearance. cus on buckling-load optimization of rectangular,
The effects of flexural orthotropy and flexural graphite-epoxy laminates that are balanced and symmet-
anisotropy on the buckling behavior of long rectangular ric.
plates that are subjected to single and combined loading Studies that address the effects of membrane
conditions are becoming better understood. For exam- orthotropy and membrane anisotropy on the buckling be-
ple, in-depth parametric studies that show the effects of havior of long rectangular plates that are restrained
flexural orthotropy and flexural anisotropy on the buck- against axial thermal expansion or contraction and sub-
ling behavior of long plates that are subjected to com- jected to uniform heating or cooling and mechanical
pression, shear, pure inplane bending, and various loads have been presented in Refs. 16 and 17. Likewise,
combinations of these loads have been presented in Refs. similar results for plates that are either fully restrained or
1 through 3. The results presented in these references de- elastically restrained against thermal expansion and con-
tail the ways in which the importance of flexural anisot- traction and subjected to uniform heating or cooling have
ropy, on the buckling resistance of long plates, varies been presented in Refs. 18 and 19, respectively. These
with the magnitude and type of the combined loading studies are comprehensive and have provided a better un-
condition. Similar results for plates loaded by uniform derstanding of the load interaction effects of balanced
shear and a general linear distribution of axial load and unbalanced, symmetrically laminated plates that are
across the plate width have also been presented in Ref. 4. subjected to mechanical loads and restrained against
Several studies have also addressed the behavior of thermal expansion or contraction.
rectangular plates that are restrained against inplane
As evidenced by the previous studies discussed,
movement- an important research area because inplane
the effects of membrane orthotropy and anisotropy, flex-
movement is typically restricted in aerospace structures
ural orthotropy and anisotropy, and the restraint of in-
by adjacent panels and stiffeners. In particular, Harris5 plane movement on the buckling behavior of rectangular
examined the effects of lateral inplane restraint on the plates that are subjected to mechanical and thermal loads
behavior of compression-loaded, specially orthotropic are becoming better understood. However, comprehen-
plates, and Obraztsov and Vasil’ev6 examined the same sive review of these studies indicates that there remains
effects on compression-loaded, balanced angle-ply lam- a need for in-depth studies that address, in a broad way,
inates. Sherbourne and Pandey7 examined the behavior the effect of a compliant, elastic restraining medium on
of balanced and unbalanced, symmetric laminates sub- the buckling of compression-loaded, symmetrically lam-
jected to uniaxial compression loads and with either lat- inated plates that are restrained against lateral expansion
eral inplane movement restrained or both lateral and contraction and inplane shearing deformations. This
movement and inplane shear deformation restrained. research area is important because it represents a class of
Their study highlights the effects of fiber orientation and problems that must be well understood in order to deter-
plate aspect ratio for selected laminate stacking sequenc- mine potential benefits and pitfalls of structural tailoring.
es made of a typical graphite-epoxy material system, and Thus, the objective of the present study is to present an
explored the possibility of tailoring laminates to have a analytical approach that indicates the effects of a compli-
negative Poisson’s ratio, that results in improved buck- ant, elastic restraining medium on the buckling behavior
ling resistance (Insight into how laminate stacking se- of compression-loaded, balanced and unbalanced, sym-
quence affects Poisson’s ratio is found in Refs. 8-10.). metrically laminated plates in a very general manner.
Similar studies that focus on buckling-load optimization Towards that objective, a buckling analysis is presented
of compression-loaded rectangular laminates are pre- first that follows the analysis presented in Ref. 19. To
sented in Refs. 11 and 12. These two studies examine the achieve this objective, the buckling analysis is formulat-
effects of unloaded edges that are either rigidly or elasti- ed in terms of nondimensional buckling coefficients and
cally restrained agained inplane, lateral movement, and load factors that depend on the inplane compliance coef-
include the effects of transverse-shear flexibility. Results ficients for a given plate and the relative compliance of a
are presented for balanced, symmetric laminates made of restraining medium. Results are then presented for infi-
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nitely long plates with the two long, unloaded edges composite-plate buckling behavior that are expressed in
clamped or simply supported and elastically restrained terms of the minimum number of independent parame-
against inplane movement. These results include nondi- ters needed to fully characterize the behavior, and that in-
menional buckling loads for selected laminates that are dicate the overall trends and sensitivity of the results to
made from one of nine different material systems, and changes in the parameters. The nondimensional param-
generic results that are applicable to a vast range of lam- eters that were used to formulate the buckling analysis
inate constructions, including hybrid laminates. Buck- are given by
ling results for infinitely long plates are important
because they often provide a practical estimate of the be- α = b D11 1/4 (1)
∞ λ D
havior of finite-length rectangular plates (lower bounds 22
to festoon buckling curves), and they provide informa- β= D12+2D66 (2)
tion that is useful in explaining the behavior of these fi- (D D )1/2
11 22
nite-length plates. Moreover, knowledge of the behavior
of infinitely long plates can provide insight into the γ= D16 (3)
buckling behavior of more complex structures such as (D131D22)1/4
stiffened panels. Finally, an example is presented that δ= D26 (4)
illustrates the use of the generic buckling-design curves (D D3)1/4
presented herein and in Ref. 18, and highlights the ef- 11 22
fects of a restraining medium on the buckling behavior. where b is the plate width and λ is the half-wave
length of the buckle pattern of an infinitely long plate
Analysis Description (see Fig. 1). The subscripted D-terms are the bending
stiffnesses of classical laminated-plate theory. The
In preparing generic design charts for buckling of a parameters α and β characterize the flexural orthot-
single flat thin plate, a special-purpose analysis is often ∞
ropy, and the parameters γ and δ characterize the flex-
preferred over a general-purpose analysis code, such as a
ural anisotropy.
finite-element code, because of the cost and effort that is
The mechanical loading conditions that are includ-
usually involved in generating a large number of results
ed in the buckling analysis are uniform transverse ten-
with a general-purpose code. The results presented in the
sion or compression, uniform shear, and a general linear
present paper were obtained by using such a special-pur-
distribution of axial load across the plate width, as de-
pose buckling analysis that is based on classical laminat-
picted in Fig. 1. Typically, an axial stress resultant dis-
ed-plate theory. The analysis details are lengthy; hence,
tribution is partitioned into a uniform part and a pure
only a brief description of the buckling analysis is pre-
bending part. However, this representation is not unique.
sented herein. First, the buckling analysis for long plates
The longitudinal stress resultant N is partitioned in the
that are subjected to a general set of mechanical loads is x
analysis into a uniform tension or compression part and
described. Then, the mechanical loads that are induced
a linearly varying part that corresponds to eccentric in-
in compression-loaded plates by elastically restraining
plane bending loads. This partitioning is given by
the inplane lateral and shear deformations are derived.
N = N –N [ε +(ε –ε )η] (5)
x xc b 0 1 0
Buckling Analysis
where N denotes the intensity of the constant-valued
Symmetrically laminated plates can have many dif- xc
tension or compression part of the load, and the term
ferent constructions because of the wide variety of mate-
containing N defines the intensity of the eccentric
rial systems, fiber orientations, and stacking sequences b
inplane bending load distribution. The symbols ε and
that can be selected to construct a laminate. A way of 0
ε define the distribution of the inplane bending load,
coping with the large number of choices for laminate 1
and the symbol η is the nondimensional coordinate
constructions is to use convenient nondimensional pa-
given by η = y/b. This particular way of partitioning
rameters in order to understand overall behavioral trends
the longitudinal stress resultant was used for conve-
and sensitivities of the structural behavior to perturba-
nience by eliminating the need to calculate the uniform
tions in laminate construction. The buckling analysis
and pure bending parts of an axial stress resultant distri-
used in the present paper is based on classical laminated-
bution prior to performing a buckling analysis.
plate theory and the classical Rayleigh-Ritz method, and
The analysis is based on a general formulation that
is derived explicitly in terms of the nondimensional pa-
includes combined destabilizing loads that are propor-
rameters defined in Refs. 1-4 and 16-20. This approach
was motivated by the need for generic (independent of a tional to a positive-valued loading parameter p that is
specific laminate construction) parametric results for increased until buckling occurs, and independent subcrit-
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ical combined loads that remain fixed at a specified load N b2
level below the value of the buckling load. Herein, the n = yj (11)
yj π2D
term "subcritical load" is defined as any load that does 22
not cause buckling to occur. In practice, the subcritical N b2
loads are applied to a plate prior to, and independent of, n = xyj (12)
xyj π2(D D3)1/4
the destabilizing loads with an intensity below that which 11 22
will cause the plate to buckle. Then, with the subcritical N b2
loads fixed, the active, destabilizing loads are applied by nbj=π2(D bjD )1/2 (13)
increasing the magnitude of the loading parameter until 11 22
buckling occurs. This approach permits certain types of where the subscript j takes on the values of 1 and 2. In
combined-load interaction to be investigated in a direct addition, the destabilizing loads are expressed in terms
and convenient manner. For example, in analyzing the of the loading parameter p in the analysis by
stability of an aircraft fuselage, the nondestabilizing
transverse tension load in a fuselage panel that is caused nxc1 =L1p (14)
by cabin pressurization can be considered to remain con-
n =L p (15)
stant and, as a result, it can be represented as a passive, y1 2
subcritical load. The combined shear, compression, and n =L p (16)
xy1 3
inplane bending loads that are caused by flight maneu-
vers can vary and cause buckling and, as a result, they n =L p (17)
b1 4
can be represented as active, destabilizing loads.
The distinction between the active, destabilizing where L1 through L4 are load factors that determine
and passive subcritical loading systems is implemented the specific form (relative contributions of the load com-
in the buckling analysis by partitioning the prebuckling ponents) of a given system of destabilizing loads. Typi-
stress resultants as follows cally, the dominant load factor is assigned a value of 1
and all others are given as positive or negative fractions.
N =–Nc +Nc (6)
xc x1 x2 Nondimensional buckling coefficients that are used
N =–N +N (7) herein are given by the values of the dimensionless stress
y y1 y2
resultants of the system of destabilizing loads at the onset
N =N +N (8) of buckling; i.e.,
xy xy1 xy2
N =N +N (9) Nc b2
b b1 b2 K ≡ nc = x1 cr =L p (18)
x x1 cr π2(D D )1/2 1 cr
where the stress resultants with the subscript 1 are the 11 22
destabilizing loads, and those with the subscript 2 are N b2
the subcritical loads. The sign convention used herein K ≡ n = y1 cr =L p (19)
y y1 cr π2D 2 cr
for positive values of these stress resultants is shown in 22
Fig. 1. In particular, positive values of the general linear N b2
edge stress distribution parameters N , N , ε , and ε K ≡ n = xy1 cr =L p (20)
b1 b2 0 1 s xy1 cr π2(D D3)1/4 3 cr
correspond to compression loads. Negative values of 11 22
N and N , or negative values of either ε or ε, yield N b2
b1 b2 0 1 K ≡ n = b1 cr =L p (21)
linearly varying stress distributions that include tension. b b1 cr π2(D D )1/2 4 cr
11 22
Depictions of a variety of inplane bending load distribu-
tions are given in Ref. 4. The two normal-stress result- where the quantities enclosed in the parentheses with
ants of the system of destabilizing loads, Nc and N , the subscript “cr” are critical values that correspond to
x1 y1
are defined to be positive-valued for compression loads.
buckling and p is the magnitude of the loading param-
This convention results in positive eigenvalues being cr
eter at buckling. Positive values of the coefficients K
used to indicate instability due to uniform compression x
and K correspond to uniform compression loads, and
loads. y
the coefficient K corresponds to uniform positive shear.
The buckling analysis includes several nondimen- s
The direction of a positive shear-stress resultant that acts
sional stress resultants associated with Eqs. (6) through
on a plate is shown in Fig. 1. The coefficient K corre-
(9). These dimensionless stress resultants are given by b
sponds to the specific inplane bending load distribution
Nc b2 defined by the selected values of the parameters ε and
nc = xj (10) 0
xj π2(D D )1/2 ε (see Fig. 1).
11 22 1
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The mathematical expression used in the variation- were obtained for isotropic and specially orthotropic
al analysis to represent the general off-center and skewed plates that are subjected to a general linear distribution of
buckle pattern is given by axial load across the plate width and compared with re-
sults that were obtained by seven different authors (see
w ( ξ,η)= ΣN (A sinπξ+B cosπξ)Φ (η) (22) Ref. 4). In every case, the agreement was good.
N m m m
m=1
Prebuckling Stress Resultants
where ξ= x/λ and η= y/b are nondimensional coor- In general, compression-loaded plates that are sym-
metrically laminated, but unbalanced, become subjected
dinates, w is the out-of-plane displacement field, and
N to a combined inplane stress state when the lateral, in-
A and B are the unknown displacement ampli- plane expansion and contraction and inplane shearing
m m
tudes. In accordance with the Rayleigh-Ritz method, deformations are restrained at the plate edges (see Fig.
the basis functions Φ ( η) are required to satisfy the 2). As the magnitude of the compression load increases,
m
kinematic boundary conditions on the plate edges at η = the induced loads increase proportionally, which can
cause premature buckling, compared to the buckling re-
0 and 1. For the simply supported plates, the basis
sistance of the corresponding unrestrained plate. These
functions used in the analysis are given by
induced mechanical loads are determined in the present
study by using the inverted membrane constitutive equa-
Φ ( η)=sinmπη (23)
m tions that are based on classical laminated-plate theory;
that is,
for values of m = 1, 2, 3, ..., N. Similarly, for the
clamped plates, the basis functions are given by
εε x = aa11aa12aa16 NNx (25a)
Φm ( η)=cos(m–1)πη–cos(m+1)πη (24) γxyy a1126a2226a2666 Nxyy
For both boundary conditions, the two long edges of a
where ε, ε, and γ are the prebuckling, inplane strains
plate are free to move in-plane, unless noted otherwise. x y xy
and the subscripted a-terms are the plate membrane
Algebraic equations that govern the buckling be-
compliance coefficients. An alternate form of this equa-
havior of infinitely long plates are obtained by substitut-
tion (see Ref. 21, p. 79) that is also used in the present
ing the series expansion for the buckling mode given by
study, that utilizes the overall laminate properties, is
Eq. (22) into a nondimensionalized form of the second
given by
variation of the total potential energy and then comput-
ing the integrals appearing in the nondimensional second ν η
variation in closed form. The resulting equations consti- 1 – yx x,xy
E E G
tute a generalized eigenvalue problem that depends on ε x y xy N /h
x ν η x
the aspect ratio of the buckle pattern λ/b (see Fig. 1) and εy = – Exy E1 Gy,xy Ny/h (25b)
γ x y xy
the nondimensional parameters and nondimensional xy ηxy,x ηxy,y 1 Nxy/h
stress resultants defined herein. The smallest eigenvalue
E E G
x y xy
of the problem corresponds to buckling and is found by
specifying a value of λ/b and solving the corresponding where h is the laminate thickness, E and E are the lam-
x y
generalized eigenvalue problem for its smallest eigen- inate moduli, G is the laminate shear moduli, ν and
xy xy
value. This process is repeated for successive values of ν are the major and minor Poisson’s ratios, respec-
yx
λ/b until the overall smallest eigenvalue is found. tively, η and η are the coefficients of mutual influ-
x,xy y,xy
Results that were obtained from the analysis de- ence of the first kind, and η and η are the
xy,x xy,y
scribed herein for uniform compression, uniform shear, coefficients of mutual influence of the second kind.
pure inplane bending (given by ε = -1 and ε = 1), and Relationships between the various constitutive terms are
0 1
various combinations of these mechanical loads have obtained by noting that the coefficient matrix of Eq.
been compared with other results for isotropic, orthotro- (25b) is symmetric. Following Ref. 21, the coefficients
pic, and anisotropic plates that were obtained by using of mutual influence are referred to herein as shear-
other analysis methods. These comparisons are dis- extension coupling coefficients.
cussed in Refs. 1-3, and in every case the results de- The effects of the elastic boundary restraints de-
scribed herein were found to be in good agreement with picted in Fig. 2a are obtained by noting that the induced
those obtained from other analyses. Likewise, results stress resultants are proportional to the strains caused by
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American Institute of Aeronautics and Astronautics
expansions or contractions and shearing deformations of Because of these definitions, R and R are referred to
2 3
the plate and the resistance provided by the restraining herein as compliance ratios. By using Eqs. (30) and
medium. Typically, the elastic resistance of the restrain- (31), Eqs. (28) and (29) become
ing medium is simulated with linear springs and ex-
pressed in terms of the corresponding spring stiffnesses. a a –a a 1+R
In the present study, the elastic resistance of a homoge- N =N 16 26 12 66 3 (32)
y xa a 1+R 1+R –a2
neous restraining medium is described approximately in 22 66 2 3 26
terms of two overall compliance coefficients of the re- and
straining medium, denoted by C and C. In particular,
2 3 a a –a a 1+R
the action of the elastic restraining medium is represent- N =N 12 26 16 22 2 (33)
xy xa a 1+R 1+R –a2
ed in a simple manner by 22 66 2 3 26
ε = - C N (26) Next, because all the subcritical loads used in the buck-
y 2 y
ling analysis are zero-valued for this problem, N = -
and x
γ = - C N (27) Nxc 1, Ny = -Ny1, and Nxy = Nxy1 (see Figs. 1 and 2). This
xy 3 xy
substitution yields
for a uniform, positive-valued set of strains in a plate.
The negative sign in Eq. (26) indicates that a positive, a a –a a 1+R
N =Nc 16 26 12 66 3 (34)
expansional strain in the y-direction is reacted by a com- y1 x1a a 1+R 1+R –a2
22 66 2 3 26
pressive stress in the y-direction, that corresponds to a
and
negative value for N. In other words, for a plate that is
y
free to deform under axial loading, a restraining medium a a –a a 1+R
would require a compressive restoring force to suppress Nxy1=–Nxc1a a12 126+R16 212+R 2–a2 (35)
the deformation. Similarly, the negative sign in Eq. (27) 22 66 2 3 26
indicates that a positive shearing strain induces, or is
Equations similar to equations (34) and (35), that
reacted by, a negative shearing-stress resultant. In the
express the induced stress resultants in terms of the
present study, the plates are presumed to be supported
overall laminate material properties, are obtained by
and loaded such that nonuniformities in the prebuckling
substituting Eqs. (26), (27), (30), and (31) into Eq. (25b)
stress field are negligible.
and then solving two of the resulting equations for N
Expressions for the induced stress resultants are ob- y
and N in terms of the applied stress resultant N.
tained by substituting Eqs. (26) and (27) into Eq. (25a) xy x
This procedure gives
and then solving two of the resulting equations for N
y
and N in terms of the applied stress resultant N. This
step gixvyes x N =Nc 1+R3 νyx+ηxy,yηx,xy (36)
y1 x1 1+R 1+R –η η
2 3 xy,y y,xy
N =N a16a26–a12 a66+C3 (28) and
y x a +C a +C –a2
and 22 2 66 3 26 Nxy 1 =Nxc1 1+1R+R21η+xR,xy+–νηyxηyη,xy (37)
2 3 xy,y y,xy
a a –a a +C
N =N 12 26 16 22 2 (29)
xy x a +C a +C –a2 Although the focus of the present study in on compres-
22 2 66 3 26
sion loaded plates (Nc >0 ), Eqs. (34) through (37)
x1
To simplify these two equations further and to provide a clearly indicate that buckling can occur under axial ten-
simple way to estimate the influence of a restraining sion loads for some laminate constructions.
medium, the compliance coefficients of the restraining Equations (34) through (37) contain three special
medium are defined as relative proportions of the plate cases of interest. First, when a plate is rigidly restrained
compliance coefficients; that is, such that ε = γ = 0, the compliance ratios R and R
y xy 2 3
are zero valued. For this case, Eqs. (34) through (37)
C = Ra (30) give
2 2 22
and C3 = R3a66 (31) NNyxc11 = AA1112 = ν1yx–+ηηxyx,yy,ηyηy,xx,yxy (38)
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and N D 1/2
L = y1 11 =
NNxcy 1 =– AA16 = η1x,–xyη+νyηxηy,xy (39) 2 Nxc1 D22
x1 11 xy,y y,xy
a a –a a 1+R D 1/2
where A , A , and A are membrane stiffnesses of 16 26 12 66 3 11 (44)
11 12 16 a a 1+R 1+R –a2 D
classical laminated-plate theory. Second, when a plate 22 66 2 3 26 22
is elastically restrained against lateral expansion or con-
N D 1/4
traction and unrestrained against inplane shearing de- L = xy1 11 =
formations such that N = 0, the compliance ratio R 3 Nc D
xy1 3 x1 22
→ ∞. This value for R means that the restraining me-
3
dium is much more compliant in shear than the plate. a a –a a 1+R D 1/4
– 12 26 16 22 2 11 (45)
For this case, Eqs. (34) through (37) give N = 0 and a a 1+R 1+R –a2 D
xy1 22 66 2 3 26 22
N ν
y1 = yx (40) For balanced laminates, a = a = 0, Eq. (45) gives L =
Nc 1+R 16 26 3
x1 2 0, and Eq. (44) becomes
where
νyx =– aa1222 = AA12AA66––AA16A2 26 (41) L2 = NNyxc11 DD1212 1/2= 1ν+yRx 2 DD1212 1/2 (46)
11 66 16
A
Equation (40) agrees with the result given in Ref. 7 for where νyx = A12 . For an isotropic plate, Eq. (46) re-
11
the even simpler case when R = 0. Third, when a plate
2 duces to
is elastically restrained against inplane shearing defor-
mations and unrestrained against lateral expansion or L = Ny1 = ν (47)
contraction such that N = 0, the compliance ratio R 2 Nc 1+R
y1 2 x1 2
→ ∞. This value for R means that the restraining
2
medium is much more compliant in lateral expansion
where ν is Poisson’s ratio of a homogeneous, isotropic
than the plate. For this case, Eqs. (34) through (37) give
material. With L = 1 and L and L defined by Eqs.
N = 0 and 1 2 3
y1 (44) and (45), the critical value of the mechanical load-
Nxy 1 = ηx,xy (42) ing parameter pcr can be calculated by using the nondi-
Nxc1 1+R3 mensional buckling analysis. Note that pcr =
p β,γ,δ,L ,L for a given set of bending boundary
where cr 2 3
conditions (e.g., simply supported and clamped edges).
a A A –A A
ηx, x y= a1666 = A1211A2622–A1621222 (43) used hFeinreailnly t,o idt eisf iinme pthoert apnret btou cmkelinntgio snt rtehsast sthtaet ea paplsroo aacph-
plies for a more sophisticated plate theory, like a first-or-
Equations (34) through (37) define a combined der transverse-shear deformation theory, because the
loading state that is induced by elastically restraining the inplane stiffness and compliance coefficients are identi-
inplane lateral and shearing deformations of a plate. The cal to those of classical laminated-plate theory. For this
buckling problem is posed by determining the load fac-
theory, p would depend also upon additional nondi-
tors L and L that appear in Eqs. (15) and (16). For an cr
2 3 mensional parameters that characterize the transverse-
applied compression load Nxc 1, the load factor L1 = 1 by shear flexibility. Thus, the only difference in the results
definition. The values for the other two load factors that for the two plate bending theories is the actual value of
are needed to completely define the prebuckling stress
p that is used in Eqs. (18)-(20), for a given problem. It
state in the nondimensional buckling analysis are ob- cr
tained by dividing Eqs. (15) and (16) by Eq. (14), with L is also important to point out that p for a long plate
1 cr
= 1, and by using Eqs. (10) - (12). This step yields does not depend on the buckle aspect ratio parameter α∞.
This fact has been shown and discussed in Refs. 1-4.
7
American Institute of Aeronautics and Astronautics
Results for Selected Laminates plies and indicate that the laterally stiff [(±45/90) ]
2 m s
Results are presented in this section that illustrate laminates exhibit the largest values of ν for a given
yx
the behavioral trends for several selected symmetrically material system; the smallest values are exhibited by the
laminated plates that are loaded by uniform axial com- axially stiff [(±45/0) ] laminates. Moreover, for the
2 m s
pression. Nine different material systems are considered [(±45/90) ] and [(±45/0) ] laminates, the largest val-
2 m s 2 m s
that include boron-aluminum, S-glass-epoxy, a typical ue of ν is obtained for the P-100/3502 pitch-epoxy and
yx
boron-epoxy, AS4/3501-6 graphite-epoxy, AS4/3502 the boron-aluminum materials, respectively. For the
graphite-epoxy, IM7/5260 graphite-bismaleimide, Kev- [(±45/0/90) ] quasi-isotropic laminates, the largest val-
m s
lar 49-epoxy, IM7/PETI-5, and P-100/3502 pitch-epoxy ue of ν is obtained for the Kevlar 49-epoxy material.
materials (see Table 1). The plates are either rigidly re- yx
For all the laminates and material systems,
strained, elastically restrained against only lateral, in-
plane movement, elastically restrained against only 0< Ny1<Nxc1 for all allowable values of the compliance
ratio R and the laminates experience a state of uniform
inplane shear deformation, or are elastically restrained 2
against both lateral inplane movement and inplane shear- biaxial compression prior to buckling.
ing deformations. First, results are presented in Table 2 Nondimensional buckling loads are shown in Fig. 3
and Figs. 3-10 for several balanced, symmetric laminates as a function of the number of laminate plies for the axi-
that exhibit relatively small degrees of flexural anisotro- ally stiff [(±45/0) ] laminates made of the IM7/5260
2 m s
py; that is, [(±45/0) ] axially stiff laminates, [(±45/ material. The nondimensional buckling load is given by
2 m s
90) ] laterally stiff laminates, [(±45/0/90) ] quasi-iso-
2 m s m s
tropic laminates, and [(±θ) ] angle-ply laminates. Then, N c b2
results are presented for [(+m4s5/0) ], [(+45/90) ], and πx12Dcr (49)
2 2 m s 2 2 m s
[(+45/0/90) ] unbalanced laminates in Tables 3-4 and
2 m s where D is defined in terms of the lamina material
Figs. 11-21 that exhibit significant degrees of membrane
properties (see Table 1) and plate thickness h by
and flexural anisotropies. In addition, results are present-
ed in Figs. 22-26 for [+θ /15] unbalanced laminates that
3 s E E h3
exhibit a wide range of values for the overall laminate D= 12 1–LνT ν (50)
Poisson’s ratio, including negative values. All the results LT TL
are based on classical laminated-plate theory and the
This bending stiffness is used herein to permit the buck-
nominal ply thickness used in the calculations was 0.005
ling performance of laminates that are made of the same
in.
material, but with different ply orientations, to be com-
pared directly. Two groups of curves are shown in the
Results for Balanced Laminates figure; the dashed and solid curves correspond to results
for clamped and simply supported plates, respectively.
For balanced laminates, there are no inplane shear-
Four curves appear within each group that correspond to
ing deformations because A = A = a = a = 0, and
16 26 16 26 values of the compliance ratio given by R = 0 (rigidly
as a result, the compliance ratio R is immaterial and the 2
3 restrained), 0.1, 0.5, and ∞. The results for R = ∞ cor-
induced shearing stress resultant N is zero valued. 2
xy1 respond to an unrestrained plate for which N = N =
With the use of Eq. (41), Eq. (36) simplifies to y1 xy1
0.
The results in Fig. 3 show a monotonic reduction in
1+ R2 NNyc1 =νyx= AA12 (48) the nondimensional buckling load as the number of plies
x1 11
(8m) increases. However, because D increases with the
where 0 ≤ R < ∞. Thus, graphs or tables of ν for cube of the plate thickness h, the actual buckling load in-
2 yx
creases by the factor m3 as the number of plies increases.
balanced, symmetric laminates yield the induced lateral
The results in Fig. 3 also show a reduction in the buck-
stress resultant N for an infinite number of different
y1
ling load as the compliance ratio decreases, for both sim-
compliance ratios.
ply supported and clamped plates, as expected.
Results are presented in Table 2 that show the load
Specifically, the unrestrained plates, that experience
N
ratio 1+ R y1 , or equivalently, ν for [(±45/0) ], uniaxial compression, exhibit the highest buckling loads
2 Nc yx 2 m s
x1 and the corresponding rigidly restrained plates, that ex-
[(±45/90) ], and [(±45/0/90) ] laminates made from perience biaxial compression, exhibit the lowest buck-
2 m s m s
one of the nine different material systems given in Table ling loads.
1. The results are independent of the number of laminate The corresponding buckling interaction curves for
8
American Institute of Aeronautics and Astronautics
the axially stiff [(±45/0) ] laminates of Fig. 3 with m ported edges is presented in Fig. 6. Each curve in this
2 m s
=1 (8 plies) and m = 6 (48 plies) are shown in Fig. 4. figure corresponds to one of the nine material systems
The dashed and solid curves correspond to results for defined in Table 1. A thick, solid gray curve is also
clamped and simply supported plates, respectively. In shown for plates made of aluminum with an elastic mod-
addition, the horizontal, flat portions of the curves corre- ulus E = 106 psi, a Poisson’s ratio ν = 0.33, and a density
spond to wide-column buckling modes for infinitely long ρ = 0.1 lb/in3. In this figure, the buckling loads are nor-
Al
plates. Points on the curves that correspond to values of malized by the bending stiffness D , which is obtained
R = 0 (rigidly restrained), 0.1, 0.5, 1, 10, and ∞ (unre- Al
2 by substituting the properties for the aluminum material
strained) are indicated by six different symbols. These into Eq. (50). Moreover, the nondimensional buckling
points are located by noting that the slope of a line that loads are weighted by the density ratio ρ /ρ, where ρ is
Al
emanates from the origin in the figure, is given by
the density of the material (see Table 1) that corresponds
to a given curve in the figure. Thus, plates with higher
Ny 1 = νyx (51) buckling resistance per unit mass are represented by
Nc 1+R curves that are farther from the origin of the graph. Also,
x1 2
points on the curves that correspond to values of R = 0
2
For these axially stiff laminates, ν = 0. 193. Addition- (rigidly restrained), 0.1, 0.5, 1, 10, and ∞ (unrestrained)
yx
ally, the 8-ply plates have a much higher degree of flex- are indicated by symbols.
ural anisotropy (γ = 0.18, δ = 0.21) than the 48-ply The results in Fig. 6 show that all the materials out
plates (γ = 0.01, δ = 0.02). These results indicate that perform the aluminum material except the Kevlar 49-ep-
the rigidly restrained plates (R = 0) experience the larg- oxy and S-glass-epoxy materials. The best performance
2
est amount of transverse compression, as expected. is exhibited by the P-100/3502 pitch-epoxy material, fol-
Buckling interaction curves for [(±45/0)], [(±45/ lowed by the boron-aluminum material. The worst per-
2 2 s
formance is exhibited by the S-glass-epoxy material,
90 )], and [(±45/0/90)] 16-ply laminates made of the
2 2 s 2 s followed by the Kevlar 49-epoxy material. The symbols
IM7/5260 material are presented in Fig. 5 for simply sup-
shown in the figure indicate a very pronounced effect of
ported (solid lines) and clamped (dashed lines) boundary
lateral edge restraint, which varies somewhat with mate-
conditions. Points on the curves for values of R = 0
2 rial system.
(rigidly restrained), 0.1, 0.5, 1, 10, and ∞ (unrestrained)
Results are presented in Fig. 7 for [(±θ) ] bal-
are also indicated by six different symbols. These points m s
are also located by noting that the slope of a line that em- anced, angle-ply laminates that dramatize the effects of
anates from the origin in the figure, is given by Eq. (51). fiber orientation and material system on the induced lat-
Values of νyx for these laminates, to be used with Eq. eral load Ny1. Each curve in this figure also corresponds
(51), are given in Table 2. Like for Fig. 4, the horizontal, to one of the nine material systems defined in Table 1,
flat portions of the curves correspond to wide-column and is independent of the number of laminate plies.
buckling modes. Moreover, the results are applicable to an infinite range
of compliance ratios given by 0 ≤ R < ∞. The results in
The results in Fig. 5 show the basic effects of ply 2
Fig. 7 show the largest variation in the load ratio
orientation on the buckling resistance of the 16-ply lam-
inates, as the laminate configuration changes from axial- Ny1 /Nxc1 in the approximate range of 50 deg < θ < 80
ly stiff to quasi-isotropic to transversely stiff. In general, deg. The greatest variations in the load ratio are exhibit-
the transversely stiff [(±45/902)2]s, plate exhibits the ed by the laminates made of the P-100/3502 pitch-epoxy
greatest buckling resistance for states of biaxial com- material, followed by those made of the IM7/PETI-5 ma-
pression. In contrast, the axially stiff [(±45/0)] plate terial. In contrast, the smallest variations are exhibited
2 2 s
exhibits the lowest buckling resistance for states of biax- by the laminates made of the boron-aluminum material,
ial compression. For a state of uniaxial compression, the followed by those made of the S-glass-epoxy material. It
quasi-isotropic laminate exhibits the greatest buckling is important to note that the induced lateral load for many
resistance. For all cases, the clamped plates are more of the laminate configurations exceeds the magnitude of
buckling resistant than the simply supported plates, as the applied load; the largest being about 3.3 times the ap-
expected. Moreover, the simply supported plates exhibit plied load.
wide-column modes for the smaller values of the compli- The effects of fiber orientation and lateral edge re-
ance ratio R2 designated by the symbols, whereas none of straint on the buckling resistance of 4-ply, highly aniso-
the clamped plates exhibit wide-column modes. tropic [±θ] laminates made of the IM7/5260 material
s
A comparison of the structural efficiency of 16-ply given in Table 1 are shown in Fig. 8. In particular, the
[(±45/0/90)] quasi-isotropic plates with simply sup- nondimensional buckling load defined by Eq. (49) is giv-
2 s
9
American Institute of Aeronautics and Astronautics
en as a function of the fiber angle θ. Two groups of extensional coupling coefficients are nonzero; that is, a
16
curves are shown in the figure. The dashed and solid ≠ 0 and a ≠ 0 (see Eq. (25a). The induced loads associ-
26
curves correspond to results for clamped and simply sup- ated with restraining lateral movement and inplane
ported plates, respectively. In addition, four curves ap- shearing deformations are given by Eqs. (34) and (35), or
pear within each group that correspond to values of the by Eqs. (36) and (37). Two special cases that will be ad-
compliance ratio given by R = 0 (rigidly restrained), dressed subsequently are the cases in which only the lat-
2
0.1, 0.5, and ∞ (unrestrained, N = N = 0). eral inplane movement of a plate is restrained (R → ∞)
y1 xy1 3
The results in Fig. 8 show a big effect of fiber ori- and the case in which only the shear deformation of a
entation and lateral edge restraint. Unlike the corre- plate is restrained (R → ∞). The induced loads for the
2
sponding results for the load ratio shown in Fig. 7, the first case are defined by Eqs. (40) and (41), and those for
largest variation in buckling load is in the approximate the second case are defined by Eqs. (42) and (43).
range of 25 deg < θ < 70 deg. Generally, the results in-
Results are presented in Table 3 that show the load
dicate that the clamped plates are more buckling resistant
than the simply supported plates, as expected, but for ratio 1 +R Ny1, or equivalently, ν for [(+45/0) ],
several values of θ, the unrestrained, simply supported 2 Nxc1 yx 2 2 m s
[(+45/90) ], and [(+45/0/90) ] unbalanced laminates
plates are more buckling resistant than the corresponding 2 2 m s 2 m s
made from one of the nine different material systems
clamped plates with R = 0 (rigidly restrained) and R =
2 2 given in Table 1 and for the case in which only the lateral
0.1. This somewhat surprising result illustrate a detri-
inplane movement of a plate is restrained (R → ∞); that
mental effect of the biaxial compression state that is in- 3
is, inplane shear deformations are unrestrained and N
duced by severely restraining the lateral movement of the xy1
clamped-plate edges. Thus, neglecting the effects of in- = 0. The results are independent of the number of lami-
plane restraint in a preliminary-design buckling analysis nate plies and indicate that the [(+45/90) ] laminates
2 2 m s
could lead to an erroneous representation of the true re- exhibit the largest values of ν for a given material sys-
yx
sponse and negative margins of safety. tem; the smallest values are exhibited by the [(+45/0) ]
2 2 m s
The results in Fig. 8 also show significant differ- laminates. Moreover, for the [(+45/90) ] laminates,
2 2 m s
ences in the shapes of the curves for the corresponding the largest value of ν is obtained for the Kevlar 49-ep-
yx
clamped and simply supported plates with R = 0, 0.1, oxy material. For the [(+45/0/90) ] and [(+45/0) ]
2 2 m s 2 2 m s
and 0.5 and in the range of approximately 25 deg < θ < laminates, the largest value of ν is obtained for the bo-
yx
65 deg. Insight into these differences is obtained by ex- ron-aluminum material. For all the laminates and materi-
amining the correponding buckling interaction curves
al systems, 0< N <Nc for all allowable values of the
that are presented in Figs. 9 and 10 for the simply sup- y1 x1
ported and clamped [±θ] laminates, respectively, made compliance ratio R2 and the laminates experience a state
s of uniform biaxial compression.
of the IM7/5260 material. Five curves are shown in Fig.
9, and in Fig. 10, that correspond to values of θ = 15, 30, Results are presented in Table 4 that show the load
45, 60, and 75 deg. Also, points on the curves that cor- ratio 1 +R Nxy1, or equivalently, η for [(+45/
respond to values of R = 0 (rigidly restrained), 0.1, 0.5, 3 Nc x,xy 2
2 x1
1, 10, and ∞ (unrestrained) are indicated by six different 0) ], [(+45/90) ], and [(+45/0/90) ] laminates made
2 m s 2 2 m s 2 m s
symbols. Comparison of the symbols shown in Fig. 9 for from one of the nine different material systems and for
the simply supported plates indicate wide-column buck- the case in which only the inplane shear deformation of
ling modes for values of θ = 30, 45, and 60 deg. For the a plate is restrained (R → ∞); that is, inplane lateral
remaining values of θ, the buckling modes are not wide- movements are unrestrai2ned and N = 0. These results,
y1
column modes. In addition, none of the modes for the
all negative, are also independent of the number of lam-
clamped plates with the values of R indicated by the
2 inate plies and indicate that the [(+45/90) ] laminates
2 2 m s
symbols are wide-column modes. Thus, the difference in exhibit the largest magnitudes of η for a given mate-
the shape of the curves in Fig. 8, for the clamped and x,xy
rial system. The smallest magnitudes of η are exhib-
simply supported plates with lateral edge restraint, ap- x,xy
ited by the [(+45/0) ] laminates. Moreover, for the
pear to be associated with the difference in mode shapes. 2 2 m s
[(+45/90) ] and [(+45/0/90) ] laminates, the largest
2 2 m s 2 m s
magnitude of η is obtained for the P-100/3502 pitch-
x,xy
Results for Unbalanced Laminates epoxy material. For the [(+45/0) ] laminate, the larg-
2 2 m s
For unbalanced laminates, inplane shearing defor- est magnitude of η is obtained for the S-glass-epoxy
x,xy
mations will develop under uniaxial compression load- material. For all the laminates and material systems, and
ing, unless rigidly restrained, because the shear- for all allowable values of the compliance ratio R, the
3
10
American Institute of Aeronautics and Astronautics